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Full text of "Treatise On Analysis Vol-Ii"

16    BAIRE'S THEOREM AND ITS CONSEQUENCES        95

20.    Let Z be a complete metric space and / a weakly continuous mapping of Z into a
separable Hilbert space E.

(a)    If A is any closed convex set in E, show that /"HA) is closed in Z (Cf. Section
12.15, Problem 3(b)).

(b)    For each integer n > 1, let Dn be the set of points z e Z such that the oscillation
of the function / (considered as a mapping of a metric space into a metric space) is
;>!/. The set Dn is closed (Section 12.7, Problem 4). Show that Dw is meager in Z.
(Cover E by a sequence (Sk) of closed balls of diameter < 1/2/z, and use (a) and Baire's
theorem to show that Dn n/-1(S*) is a nowhere dense closed set in Z, for each k.)

(c)    Deduce from (a) that there exists a meager set M in Z such that / is strongly
continuous at every point of Z  M.

21.    Let E = JT(R) be the subspace consisting of continuous functions with compact
support, in the Banach space ^(R) of real-valued bounded functions on R. Let u

n

be the continuous linear form /(-/(/:) on E. Show that the sequence (un) converges

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simply on E but is not bounded in the dual E', and that its limit u is not continuous
on E.

22.    Let E be the subspace consisting of sequences () with finite support, in the Hilbert
space /2, so that E is the set of all (finite) linear combinations of the vectors en in I2
(6.5). Let u be the endomorphism of E for which u(en) = nen for all n. Show that, for
every y e E, the linear form x\- *(u(x)\y) is continuous on E, but that u is not a con-
tinuous endomorphism of E.

23.    Let E be the subset of R2 consisting of the line D = {0} x R and the points (I In, kin2),
where n and k belong to Z and n > 0.

(a)    For every point (0, y) e E and every integer n > 0, let Tn(y) be the set of points
(u, u) e E such that u^l/n and \v  y\<Zu. Show that if we take the sets Tn(y) (n^l)
as a fundamental system of neighborhoods of (0, y), for each y e R, and if for each
other point z of E we take the set consisting of that point alone as a fundamental
system of neighborhoods of z, then we have a HausdorfT topology 9~ on E, for which
each subspace Tn(y) is metrizable and compact. The topology induced by F on D is
the discrete topology.

(b)    Let A be the set {0} x Q, which is closed in D and whose complement B in D
is also closed, If a set U which is open with respect to & contains B, show that there
exists an interval {0} x [a, b] in D (with a < b) and an integer n such that U contains
the union of the sets Tn(y) for y e [a, b] and (0, y) e B. (Use the fact that Q is a
meager subset of R in the usual topology.) Deduce that every neighborhood of A in E
meets U, and hence that the space E is not metrizable, although it contains a discrete
denumerable dense subset (use (4.5.2)).

24.    Let (an) be a sequence of real numbers. Suppose that, for each sequence () of real
numbers belonging to the space /* (Section 5.7, Problem 1) (resp. to the space /2
(6.5)), the series with general term ann is convergent. Prove that the sequence (#)
belongs to / (Section 5.6, Problem 1) (resp, belongs to /2). (For each integer N,

N

consider the continuous linear form x i > JT an $ n on /* (resp. /2), and apply the Banach-

irl

Steinhaus theorem.) and a